Modeling and Replication of the Prepayment Option of Mortgages including Behavioral Uncertainty

ArXiv ID: 2410.21110 “View on arXiv”

Authors: Unknown

Abstract

Prepayment risk embedded in fixed-rate mortgages forms a significant fraction of a financial institution’s exposure, and it receives particular attention because of the magnitude of the underlying market. The embedded prepayment option (EPO) bears the same interest rate risk as an exotic interest rate swap (IRS) with a suitable stochastic notional. We investigate the effect of relaxing the assumption of a deterministic relationship between the market interest rate incentive and the prepayment rate. A non-hedgeable risk factor is modeled to capture the uncertainty in mortgage owners’ behavior, leading to an incomplete market. We prove under natural assumptions that including behavioral uncertainty reduces the exposure’s value. We statically replicate the exposure resulting from the EPO with IRSs and swaptions, and we show that a replication based on swaps solely cannot easily control the right tail of the exposure distribution, while including swaptions enables that. The replication framework is flexible and focuses on different regions in the exposure distribution. Since a non-hedgeable risk factor entails the existence of multiple equivalent martingale measures, pricing and optimal replication are not unique. We investigate the effect of a market price of risk misspecification and we provide a methodology to generate robust hedging strategies. Such strategies, obtained as solutions to a saddle-point problem, allow us to bound the exposure against a misspecification of the pricing measure.

Keywords: prepayment risk, interest rate swaps, swaptions, replication, incomplete markets, Mortgages / Fixed Income Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper involves advanced mathematical modeling of incomplete markets and robust hedging via saddle-point problems, but lacks empirical validation (e.g., backtests, datasets, or statistical metrics) and focuses on theoretical replication frameworks.

  flowchart TD
    A["Research Goal:<br>Model & replicate EPO<br>with behavioral uncertainty"] --> B["Methodology:<br>Stochastic model for<br>behavioral uncertainty"]
    
    B --> C["Data/Inputs:<br>Market rates, mortgage features,<br>behavioral parameters"]
    
    C --> D["Computational Process:<br>Solve incomplete market model<br>under multiple martingale measures"]
    
    D --> E["Computational Process:<br>Static replication using<br>IRSs and Swaptions"]
    
    E --> F["Outcome 1:<br>Behavioral uncertainty<br>reduces exposure value"]
    
    E --> G["Outcome 2:<br>IRS-only replication fails<br>to control right tail risk"]
    
    E --> H["Outcome 3:<br>Swaptions enable robust<br>tail risk management"]
    
    E --> I["Outcome 4:<br>Robust hedging via<br>saddle-point optimization"]